Thinking Mathematically (6th Edition)

a. $\sim p\wedge q$ b. $\begin{array}{lllll} p & q & \sim p & \sim p\wedge q & \\ \hline T & T & F & F & \\ T & F & F & F & \\ F & T & T & T & \\ F & F & T & F & \end{array}$ c. when p is false, and q is true.
Restructuring the statement: NOT( you did the dishes) AND (you left the room in a mess) a. p: You did the dishes; q: You left the room in mess $\sim p\wedge q$ b. Set up a truth table for two inputs, p and q: $\begin{array}{llll} p & q & ... & ...\\ \hline T & T & & \\ T & F & & \\ F & T & & \\ F & F & & \end{array}$ Next column: $\sim p$, negation of column 1, $\begin{array}{lllll} p & q & \sim p & & \\ \hline T & T & F & & \\ T & F & F & & \\ F & T & T & & \\ F & F & T & & \end{array}$ Next column: $\sim p\wedge q$, conjunction of columns 3 and 2 $\begin{array}{lllll} p & q & \sim p & \sim p\wedge q & \\ \hline T & T & F & F & \\ T & F & F & F & \\ F & T & T & T & \\ F & F & T & F & \end{array}$ c. The statement is true (see row 3) when p is false, and q is true.